The God has created a man in order that he creates that the God fails to do

Thursday, 5 May 2011

What is a discrete space-time?

A concept of discrete space-time was first suggested by V. Ambartsumian and D. Ivanenko in 1930. This concept has not been formulated in a strict mathematical form, but often is related to a hypothesis of a minimal fundamental length, e. g., in non-linear field theories with self-interaction constants of dimension of length. One usually considers a lattice in coordinate space and introduces a cut-off factor in a momentum space.

An acceptable mathematical formalization of a discrete space-time can be given in terms of topological spaces Y where a connected component of any point y of Y is the closure of y. For instance, if Y is a Hausdorff space, then a connected component of its point y is this point itself,  i. e., Y is a totally disconnected space.

This is the case of a discrete topological space, the space of rational numbers, vector spaces and analytic manifolds over fields with ultra-metric absolute values. In algebraic quantum field theory, the spectrum of some C*-algebras, e. g., the C*-algebras of probability measures, is totally disconnected.


G. Sardanashvily, Discrete space-time, Enciclopaedia of Mathematics (Springer)

D.Ivanenko, G.Sardanashvily, Towards a model of discrete space-time, Russ. Phys. J. 21 (1978) 1508.

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